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Z-Scores Sometimes we want to do more than summarize a bunch of scores. Sometimes we want to talk about particular scores within the bunch. We may want to tell other people about whether or not a score is above or below average. We may want to tell other people how far away a particular score is from average. We might also want to compare scores from different bunches of data. We will want to know which score is better. Z-scores can help with all of this! They Tell Us Important Things Z-Scores tell us whether a particular score is equal to the mean, below the mean or above the mean of a bunch of scores. They can also tell us how far a particular score is away from the mean. Is a particular score close to the mean or far away? If a Z-Score…. Has a value of 0, it is equal to the group mean. Is positive, it is above the group mean. o Is equal to +1, it is 1 Standard Deviation above the mean. o Is equal to +2, it is 2 Standard Deviations above the mean. Is negative, it is below the group mean. o Is equal to -1, it is 1 Standard Deviation below the mean. o Is equal to -2, it is 2 Standard Deviations below the mean. Z-Scores Can Help Us Understand… How typical a particular score is within bunch of scores. If data are normally distributed, approximately 95% of the data should have Z-scores between -2 and +2. Z-scores that do not fall within this range may be less typical of the data in a bunch of scores. Z-Scores Can Help Us Compare… Individual scores from different bunches of data. We can use Z-scores to standardize scores from different groups of data. Then we can compare raw scores from different bunches of data. Chapter 2: The Normal Distributions Key Vocabulary: mu Density curve Inflection point Normal distribution Standard Normal z-scores sigma Empirical rule distribution Outcomes Percentile Normal Probability plot Simulation N Normal curve Standardized value normalcdf invNorm 2.1 , Density Curves and Measures of Relative Standing: 1. Explain the two ways to describe the position of Jenny’s score on the first statistics test within the distribution of test scores. 2. How do we calculate the percent of observations falling within k standard deviations of the mean? 3. List the 4 steps used to explore data from a single quantitative variable. 4. Describe the relationship between the mean and the median of a skewed distribution. 2.2 Normal Distributions: 5. List the three reasons that normal distributions are important in statistics. 6. Give 3 examples of distributions that are often close to normal. 7. How is a standard normal distribution different from a normal distribution? 8. List the steps used in solving problems involving normal distributions. 9. In what situation do we use Table A backwards? 10. Is there a difference between the 80th percentile and the top 80%? Explain 11. Explain the basic idea of a Normal probability plot. 12. How does a Normal probability plot indicate that the data are normal?